title: Convex reformulation of biologically-based multi-criteria intensity-modulated radiation therapy optimization including fractionation effects.
abstract: Finding fluence maps for intensity-modulated radiation therapy (IMRT) can be formulated as a multi-criteria optimization problem for which Pareto optimal treatment plans exist. To account for the dose-per-fraction effect of fractionated IMRT, it is desirable to exploit radiobiological treatment plan evaluation criteria based on the linear-quadratic (LQ) cell survival model as a means to balance the radiation benefits and risks in terms of biologic response. Unfortunately, the LQ-model-based radiobiological criteria are nonconvex functions, which make the optimization problem hard to solve. We apply the framework proposed by Romeijn et al (2004 Phys. Med. Biol. 49 1991-2013) to find transformations of LQ-model-based radiobiological functions and establish conditions under which transformed functions result in equivalent convex criteria that do not change the set of Pareto optimal treatment plans. The functions analysed are: the LQ-Poisson-based model for tumour control probability (TCP) with and without inter-patient heterogeneity in radiation sensitivity, the LQ-Poisson-based relative seriality s-model for normal tissue complication probability (NTCP), the equivalent uniform dose (EUD) under the LQ-Poisson model and the fractionation-corrected Probit-based model for NTCP according to Lyman, Kutcher and Burman. These functions differ from those analysed before in that they cannot be decomposed into elementary EUD or generalized-EUD functions. In addition, we show that applying increasing and concave transformations to the convexified functions is beneficial for the piecewise approximation of the Pareto efficient frontier.
authors.affiliation: Department of Radiation Oncology, Radboud University Nijmegen Medical Centre, PO Box 9101, 6500 HB Nijmegen, The Netherlands. a.hoffmann@rther.umcn.nl
authors.forename: Aswin L
authors.initials: AL
authors.lastname: Hoffmann
authors.forename: Dick
authors.initials: D
authors.lastname: den Hertog
authors.forename: Alex Y D
authors.initials: AY
authors.lastname: Siem
authors.forename: Johannes H A M
authors.initials: JH
authors.lastname: Kaanders
authors.forename: Henk
authors.initials: H
authors.lastname: Huizenga
mesh_terms: D002470:Cell Survival / Q000528:radiation effects; D019583:Dose Fractionation, Radiation; D006801:Humans; D016014:Linear Models; D008954:Models, Biological; D009369:Neoplasms / Q000473:pathology / Q000532:radiotherapy; D011336:Probability; D011879:Radiotherapy Dosage; D050397:Radiotherapy, Intensity-Modulated / Q000379:methods*